Properties of Functions
(2,10)
(-2 6)
(5,0)
10
5
(-8,-4)
f is increasing on the intervals(-8,-2), (0,10), ( 0 ,  )
f is decreasing on the intervals (  ,  8 ) (-2,0) (2,5)
There are local minima at f(0)= 0 and f(5)= 0
The local minima are (0, 0) and (5,0)
There are local maxima at f(-2) =6 and f(2) =10
The local maxima are (-2,6) and (2,10)
(0,0)
5
10
Library of Functions
Linear Function
The graph of a linear function is a non-vertical line. A linear function is of the form
y = f(x) = mx + b
where m and b are real numbers.
Here m is the slope and b is the y - intercept.
To find the x - intercept let y = 0 and solve for x
The domain and range of a linear function are all real numbers.
Graph
•Example
f(x) = 2x+ 3
The slope is m = 2
The y –intercept is 3
The x – intercept is –3/2
Since the slope is positive,
f is an increasing function
3

2
The domain and the range
are the set of real numbers
Example
f (x) = 4 - 7x
The slope is m = -7
The y –intercept is 4
The x – intercept is 7/4
Since the slope is negative, f is an decreasing function
The domain and the range are all real numbers
The constant function
f (x) = C
The domain of the constant function is all real numbers
The range is the constant C. In this function is equal to 3
The graph is a horizontal line
The slope is m = 0
The y – intercept is (0, 3)
Identity Function
f(x) = x
f is called the identity function because the value of y is always identical to that of x
f is a linear function because its graph is a straight line.
The domain and range are all real numbers
The x and y intercepts are both 0
The identity function divides the first and third quadrants into 45 degree angles
y y  0 
The Square function
f ( x)  x
2
The domain of the square function is all real numbers
The range of f(x) is  y
y  0
The x and y intercepts are (0,0)
The square function is even. Thus it is symmetric with the origin
The Cube Function
f ( x)  x
3
The domain and the range are all real numbers
The x and y intercepts are at (0,0)
The cube function is an odd function and thus is symmetric
with the origin
The square
function
f ( x )  root
x
The domain is
The range is
x x  0 
y y  0 
The Cube root function
f ( x) 
3
x
The domain and range are all real numbers
The cube root function is an odd function and is symmetric
with the origin
The Absolute Value Function
The domain of the absolute value function is all real numbers.
The range is all non negative numbers
The absolute value function is even and thus symmetric with the origin
The Reciprocal Function
f ( x) 
1
x
The domain is all real numbers except x = 0
The range is all real numbers except y= 0
The reciprocal function is an odd function and is thus symmetric with
the origin